On Polyakov's notion of regular q-concave CR manifolds

Let M be a C∞-submanifold of codimension k in Cn , 1 ≤ k ≤ n − 1. For a ∈ M , a collection ρ = (U ; ρ1, . . . , ρk) will be called a defining collection of M at a if U ⊆ Cn is open, a ∈ U , ρ1, . . . , ρk are real C∞-functions on U , U ∩ M = { ζ ∈ U ∣ ρ1(ζ ) = . . . ρk(ζ ) = 0 } = ∅ and dρ1(ζ ) ∧ . . . ∧ dρk(ζ ) = 0 for all ζ ∈ U ∩ M . For a ∈ M , let T a (M) be the complex tangent space of M at a. We assume that M is a generic C R-submanifold of Cn which means, by definition, that